Testing the Rank and Definiteness of Estimated Matrices with Applications to Factor, State-Space and ARMA Models

Consider any consistent, asymptotically normal estimate ǎ of an arbitrary rectangular or square matrix A. This article derives an explicit test for the rank of A and a related test of (semi) definiteness of A. Potential applications include testing for identification of structural models, testing for the number of state variables in state-space models (including tests for the order of autoregressive moving average (ARMA) processes), consumer demand analysis applications, and testing for the number of factors in factor analysis and related procedures. The test is based on the Gaussian elimination Lower-Diagonal-Upper triangular (LDU) decomposition. The test is illustrated with an empirical application to testing the order of ARMA processes.